$\begin{aligned} &F(x)=|4x-20| \\\\ &f(x)=F'(x) \end{aligned}$ $\int_{-5}^{5} f(x)\,dx=$
$f$ is the derivative of $F$, which means $F$ is an antiderivative of $f$. Since we know the antiderivative of $f$, we can use the fundamental theorem of calculus: For every function $f$ and its antiderivative $F$, $\int_a^b f(x)\,dx=F(b)-F(a)$. $\begin{aligned} &\phantom{=}\int_{-5}^{5} f(x)\,dx \\\\ &=F({5})-F({-5}) \\\\ &=|4{(5)}-20|-|4{(-5)}-20| \\\\ &=0-40 \\\\ &=-40 \end{aligned}$ In conclusion, $\int_{-5}^{5} f(x)\,dx=-40$